Wow, if my parents really wanted to know my password for anything, they could just ask. Tell your mother to stop using very intrusive options of getting your personal information. She's your mother, just say "Mom, if you really wanted to know, you could just ask."
Or, she may just be playing mind games with you so that you are more careful about which sites you visit.
But definitely, run AVG 8.5 Free. It won't scan for rootkits (in 8 it did, but Avast are a bunch of handicapped jerks), but it will scan for keyloggers.